By \(\beta > 0\)  and \(\alpha -\beta > 0\) , we apply the formula (1.100) (p.25)
in [1]}, which can be also directly verified by the expansion of
the power series of \(E_{\alpha ,\alpha -\beta }(-\lambda s^{\alpha })\) ,
so that
\(\frac{1}{\Gamma (\beta )}\int ^t_0 (t-s)^{\beta -1}s^{\alpha -\beta -1}E_{\alpha ,\alpha -\beta }(-\lambda s^{\alpha })ds= t^{\alpha -1}E_{\alpha ,\alpha }(-\lambda t^{\alpha }).\) 
